Is ${994885}$ divisible by $3$ ?
Solution: A number is divisible by $3$ if the sum of its digits is divisible by $3$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {994885}= &&{9}\cdot100000+ \\&&{9}\cdot10000+ \\&&{4}\cdot1000+ \\&&{8}\cdot100+ \\&&{8}\cdot10+ \\&&{5}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {994885}= &&{9}(99999+1)+ \\&&{9}(9999+1)+ \\&&{4}(999+1)+ \\&&{8}(99+1)+ \\&&{8}(9+1)+ \\&&{5} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {994885}= &&\gray{9\cdot99999}+ \\&&\gray{9\cdot9999}+ \\&&\gray{4\cdot999}+ \\&&\gray{8\cdot99}+ \\&&\gray{8\cdot9}+ \\&& {9}+{9}+{4}+{8}+{8}+{5} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $3$ , so the first five terms must all be multiples of $3$ That means that to figure out whether the original number is divisible by $3 $ , all we need to do is add up the digits and see if the sum is divisible by $3$ . In other words, ${994885}$ is divisible by $3$ if ${ 9}+{9}+{4}+{8}+{8}+{5}$ is divisible by $3$ Add the digits of ${994885}$ $ {9}+{9}+{4}+{8}+{8}+{5} = {43} $ If ${43}$ is divisible by $3$ , then ${994885}$ must also be divisible by $3$ Add the digits of ${43}$ $ {4}+{3} = \color{#9D38BD}{7} $ If $\color{#9D38BD}{7}$ is divisible by $3$ , then ${43}$ must also be divisible by $3$ $\color{#9D38BD}{7}$ is not divisible by $3$, therefore ${994885}$ must not be divisible by $3$.